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Zargon Glasses

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Here's a solution from Matthew (Nimbin Central School, Australia). He found that 252 lenses would be needed to make all the glasses. He explained how he did it by first working out all the combinations of glasses that made up the 9 orders:

"I wrote down in a grid that was eight wide and nine down, 3 or 4 and we had to be careful not to get the same number of threes or fours as another group of eight. And at the end we found out that there were 36 fours and 36 threes, and when we added all of the numbers together it equalled 252".

Is there another way to do the maths in this problem?